Equivalent Definitions of A Closed Set
In a topological space, a set is closed if and only if it coincides with its closure. Equivalently, a set is closed if and only if it contains all of its limit points.
This is not to be confused with a closed manifold.
Read more about this topic: Closed Set
Famous quotes containing the words equivalent, definitions, closed and/or set:
“To place oneself in the position of God is painful: being God is equivalent to being tortured. For being God means that one is in harmony with all that is, including the worst. The existence of the worst evils is unimaginable unless God willed them.”
—Georges Bataille (1897–1962)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)
“With two sons born eighteen months apart, I operated mainly on automatic pilot through the ceaseless activity of their early childhood. I remember opening the refrigerator late one night and finding a roll of aluminum foil next to a pair of small red tennies. Certain that I was responsible for the refrigerated shoes, I quickly closed the door and ran upstairs to make sure I had put the babies in their cribs instead of the linen closet.”
—Mary Kay Blakely (20th century)
“Although knaves win in every political struggle, although society seems to be delivered over from the hands of one set of criminals into the hands of another set of criminals, as fast as the government is changed, and the march of civilization is a train of felonies, yet, general ends are somehow answered.”
—Ralph Waldo Emerson (1803–1882)